Theorem 0nnq 10993

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5734	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10981	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4105	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 4004	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2108  ∀wral 3067  ∅c0 4352   class class class wbr 5166   × cxp 5698  ‘cfv 6573  2^nd c2nd 8029  Ncnpi 10913   <_N clti 10916   ~_Q ceq 10920  Qcnq 10921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-nq 10981

This theorem is referenced by:  adderpq  11025  mulerpq  11026  addassnq  11027  mulassnq  11028  distrnq  11030  recmulnq  11033  recclnq  11035  ltanq  11040  ltmnq  11041  ltexnq  11044  nsmallnq  11046  ltbtwnnq  11047  ltrnq  11048  prlem934  11102  ltaddpr  11103  ltexprlem2  11106  ltexprlem3  11107  ltexprlem4  11108  ltexprlem6  11110  ltexprlem7  11111  prlem936  11116  reclem2pr  11117